In testing the statistical significance of the relationship between two quantitative variables we will use the five step hypothesis testing procedure:
In order to use Pearson's \(r\) both variables must be quantitative and the relationship between \(x\) and \(y\) must be linear
Research Question | Is the correlation in the population different from 0? | Is the correlation in the population positive? | Is the correlation in the population negative? |
|---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\rho=0\) | \(\rho= 0\) | \(\rho = 0\) |
Alternative Hypothesis, \(H_{a}\) | \(\rho \neq 0\) | \(\rho > 0\) | \(\rho< 0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Minitab will not provide the test statistic for correlation. It will provide the sample statistic, \(r\), along with the p-value (for step 3).
Optional: If you are conducting a test by hand, a \(t\) test statistic is computed in step 2 using the following formula:
\(t=\dfrac{r- \rho_{0}}{\sqrt{\dfrac{1-r^2}{n-2}}} \)
In step 3, a \(t\) distribution with \(df=n-2\) is used to obtain the p-value.
Minitab will give you the p-value for a two-tailed test (i.e., \(H_a: \rho \neq 0\)). If you are conducting a one-tailed test you will need to divide the p-value in the output by 2.
If \(p \leq \alpha\) reject the null hypothesis, there is convincing evidence of a relationship in the population.
If \(p>\alpha\) fail to reject the null hypothesis, there is not enough evidence of a relationship in the population.
Based on your decision in Step 4, write a conclusion in terms of the original research question.